Sets: Naive, Axiomatic and Applied (eBook)

A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students

D. van Dalen, H. C. Doets, H. De Swart (Autoren)

I. N. Sneddon (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
360 Seiten
Elsevier Science (Verlag)
978-1-4831-5039-0 (ISBN)

Infos

Sets: Naïve, Axiomatic and Applied is a basic compendium on naïve, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and relative complement as well as the reflection principle, measurable cardinals, and models of set theory. Applications of the axiom of choice are also discussed, along with infinite games and the axiom of determinateness. Comprised of three chapters, this volume begins with an overview of naïve set theory and some important sets and notations. The equality of sets, subsets, and ordered pairs are considered, together with equivalence relations and real numbers. The next chapter is devoted to axiomatic set theory and discusses the axiom of regularity, induction and recursion, and ordinal and cardinal numbers. In the final chapter, applications of set theory are reviewed, paying particular attention to filters, Boolean algebra, and inductive definitions together with trees and the Borel hierarchy. This book is intended for non-logicians, students, and working and teaching mathematicians.

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Sets: Naive, Axiomatic and Applied is a basic compendium on naive, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and relative complement as well as the reflection principle, measurable cardinals, and models of set theory. Applications of the axiom of choice are also discussed, along with infinite games and the axiom of determinateness. Comprised of three chapters, this volume begins with an overview of naive set theory and some important sets and notations. The equality of sets, subsets, and ordered pairs are considered, together with equivalence relations and real numbers. The next chapter is devoted to axiomatic set theory and discusses the axiom of regularity, induction and recursion, and ordinal and cardinal numbers. In the final chapter, applications of set theory are reviewed, paying particular attention to filters, Boolean algebra, and inductive definitions together with trees and the Borel hierarchy. This book is intended for non-logicians, students, and working and teaching mathematicians.

Inhalt

Front Cover 1
Sets: Naïve, Axiomatic and Applied 4
Copyright Page 5
Table of Contents 6
Preface 10
Acknowledgements 14
Introduction 16
Chapter 1. Naive Set Theory 20
1. Some important sets and notations 20
2. Equality of sets 22
3. Subsets 23
4. The Naive Comprehension principle and the empty set 25
5. Union, intersection and relative complement Complement, de Morgan's laws 28
6. Power set 37
7. Unions and intersections of families 40
8. Ordered pairs 48
9. Cartesian product 51
10. Relations 55
11. Equivalence relations 60
12. Real numbers 70
13. Functions (mappings) 75
14. Orderings 97
15. Equivalence (cardinality) 111
16. Finite and infinite 128
17. Denumerable sets 135
18. Uncountable sets 146
19. The paradoxes 151
20. The set theory of Zermelo-Fraenkel (ZF) 155
21. Peano's arithmetic 167
Chapter 2. Axiomatic Set Theory 171
1. The axiom of regularity 171
2. Induction and Recursion 175
3. Ordinal numbers 184
4. The cumulative hierarchy 187
5. Ordinal arithmetic 193
6. Normal operations 199
7. The reflection principle 206
8. Initial numbers 209
9. The axiom of choice 213
10. Cardinal numbers 223
11. Models 235
12. Measurable cardinals 248
Chapter 3. Applications 258
1. Filters 258
2. Boolean algebra 261
3. Order types 272
4. Inductive definitions 281
5. Applications of the axiom of choice 287
6. The Borel hierarchy 294
7. Trees 316
8. The axiom, of Determinateness (AD) 332
Appendix 342
Symbols 347
Literature 350
Index 352
Other Titles in the Series 362

Produkt-Details

• Erscheint lt. Verlag: 9.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• ISBN-10: 1-4831-5039-9 / 1483150399
• ISBN-13: 978-1-4831-5039-0 / 9781483150390